You can easily get a 30-day temporary tourist licence at the Traffic Police Motor Vehicle Service Station in Beijing airport terminal 3 (and probably at other traffic bureaus depending on your point of entry into China).
You will need to bring your international driver's licence, a translation of that licence into Chinese, 3 passport pictures (35x25mm), a hotel registration (unless you've just arrived) and a provisional driving permit application form.
Chinese translation form: http://motorhome-china.com/images/2012_05_Drivers_License_Translation_Form.pdf
Permit application form: http://motorhome-china.com/images/2012_05_Provisional_Driving_Permit_Application_Form.pdf
On the day you might get someone that speaks good English, or you might not. It's luck of the draw.
You must get an eye test which costs about 10 RMB. There's a place on the second floor called The First Aid Station.
Then you need to go back down and pay another 10 RMB and you will be issued with a laminated temporary licence.
However, if you do want to hire a car and driver there are many places that offer it. Just search Google and you'll find companies like this http://www.shanghaidriver.net/en/index.php.
Look around and you can find daily rates from around 400 yuan, or up to 800 yuan for someone with a good clean, modern car.